Best Known (19, 23, s)-Nets in Base 2
(19, 23, 2049)-Net over F2 — Constructive and digital
Digital (19, 23, 2049)-net over F2, using
(19, 23, 2060)-Net over F2 — Digital
Digital (19, 23, 2060)-net over F2, using
- net defined by OOA [i] based on linear OOA(223, 2060, F2, 4, 4) (dual of [(2060, 4), 8217, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(223, 2060, F2, 3, 4) (dual of [(2060, 3), 6157, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(223, 2060, F2, 4) (dual of [2060, 2037, 5]-code), using
- 1 times truncation [i] based on linear OA(224, 2061, F2, 5) (dual of [2061, 2037, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(223, 2048, F2, 5) (dual of [2048, 2025, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(212, 2048, F2, 3) (dual of [2048, 2036, 4]-code or 2048-cap in PG(11,2)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(212, 13, F2, 11) (dual of [13, 1, 12]-code), using
- strength reduction [i] based on linear OA(212, 13, F2, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,2)), using
- dual of repetition code with length 13 [i]
- strength reduction [i] based on linear OA(212, 13, F2, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,2)), using
- linear OA(21, 13, F2, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- 1 times truncation [i] based on linear OA(224, 2061, F2, 5) (dual of [2061, 2037, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(223, 2060, F2, 4) (dual of [2060, 2037, 5]-code), using
- appending kth column [i] based on linear OOA(223, 2060, F2, 3, 4) (dual of [(2060, 3), 6157, 5]-NRT-code), using
(19, 23, 4093)-Net in Base 2 — Upper bound on s
There is no (19, 23, 4094)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 8 390654 > 223 [i]